Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-09T01:30:31.620Z Has data issue: false hasContentIssue false
  • English
  • Français

Variance asymptotics and central limit theorems for volumes of unions of random closed sets

Part of: Limit theorems Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Tomasz Schreiber
Tomasz Schreiber*
Affiliation:
Nicholas Copernicus University, Toruń
*
∗ Postal address: Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Ul Chopina 12/18, 87-100 Toruń, Poland. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X, X1, X2, … be a sequence of i.i.d. random closed subsets of a certain locally compact, Hausdorff and separable base space E. For a fixed normalised Borel measure μ on E, we investigate the behaviour of random variables μ(E \ (X1 ∪ ∙ ∙ ∙ ∪ Xn)) for large n. The results obtained include a description of variance asymptotics, strong law of large numbers and a central limit theorem. As an example we give an application of the developed methods for asymptotic analysis of the mean width of convex hulls generated by uniform samples from a multidimensional ball. Another example deals with unions of random balls in ℝd with centres distributed according to a spherically-symmetric heavy-tailed law.

Keywords

Random setscentral limit theoremvariance asymptoticsconvex hullmean width

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60F05: Central limit and other weak theorems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 3 , September 2002 , pp. 520 - 539
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baldi, P. and Rinott, Y. (1989). On normal approximations of distributions in terms of dependency graphs. Ann. Prob. 17, 1646–1650.Google Scholar
Bräker, H. and Hsing, T. (1998). On the area and perimeter of a random convex hull in a bounded convex set. Prob. Theory Relat. Fields 111, 517–550.Google Scholar
Cabo, A. J. and Groeneboom, P. (1994). Limit theorems for functionals of convex hulls. Prob. Theory Relat. Fields 100, 31–55.Google Scholar
Efron, B. (1965). The convex hull of a random set of points. Biometrika 52, 331–343.CrossRefGoogle Scholar
Groeneboom, P. (1988). Limit theorems for convex hulls. Prob. Theory Relat. Fields 79, 327–368.CrossRefGoogle Scholar
Hsing, T. (1994). On the asymptotic distribution of the area outside a random convex hull in a disk. Ann. Appl. Prob. 4, 478–493.Google Scholar
Hueter, I. (1999). Limit theorems for the convex hull of random points in higher dimensions. Trans. Amer. Math. Soc. 11, 4337–4363.CrossRefGoogle Scholar
Kendall, W. S., Mecke, J. and Stoyan, D. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Khamdamov, I. M. (1996). Limit theorems for the area and the perimeter of convex hull generated by truncated distribution. Dokl. Akad. Nauk Resp. Uzb. 5, 3–5 (in Russian).Google Scholar
Khamdamov, I. M. and Nagaev, A. V. (1991). Limiting distributions for functionals of the convex hull generated by uniformly distributed variables. Dokl. Akad. Nauk UzSSR 7, 8–9 (in Russian).Google Scholar
Küfer, K. H. (1994). On the approximation of a ball by random polytopes. Adv. Appl. Prob. 26, 876–892.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Molchanov, I. S. (1993). Limit Theorems for Unions of Random Closed Sets (Lecture Notes Math. 1561). Springer, Berlin.Google Scholar
Molchanov, I. S. (1995). On the convergence of random processes generated by polyhedral approximation of convex compacts. Theory Prob. Appl. 40, 383–390.Google Scholar
Reiss, R.-D. (1993). A Course on Point Processes. Springer, Berlin.Google Scholar
Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von n zufällig gewählten Punkten I. Z. Wahrscheinlichkeitsth. 2, 75–84.CrossRefGoogle Scholar
Rényi, A. and Sulanke, R. (1964). Über die konvexe Hülle von n zufällig gewählten Punkten II. Z. Wahrscheinlichkeitsth. 3, 138–147.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.Google Scholar
Schneider, R. (1988). Random approximation of convex sets. J. Microscopy 151, 211–227.Google Scholar
Schneider, R. (1993). Convex Bodies: the Brunn–Minkowski Theory (Encyclopaedia Math. Appl. 44). Cambridge University Press.Google Scholar
Schreiber, T. (2002). Limit theorems for certain functionals of unions of random closed sets. Theory Prob. Appl. 47, 130–142.Google Scholar